Professional Help Needed--Elementary Number Theory

I will preface this by saying that I have no formal training in Mathematics. I've taken Calc 1 and a couple of Symbolic Logic classes. Forgive me if I butchered any terminology. However, this has been bugging me for a while, and I would like to know if anyone knows anything about this.

There seems to be a predictable pattern in the gaps between successive terms in series with prime exponents. This has been known for x^2 for a long time.


x^1== +1
x^2 == 2x+1

x^3 == [(x^2+x)(3)] +1

1+ (1^2+1) (3) +1 = 8
8 + (2^2+2) (3) +1 = 27
27 + (3^2+3) (3) +1= 64 ...

x^5== [(x^2+x)(x^2+x+1)(5)]+1

1+[(1^2+1)(1^2+1+1)(5)]+1=32
32 + (2^2+2)(2^2+2+1)(5)+1=243
243+(3^2+3)(3^2+3+1)(5)+1=1024...


x^7 = {(x^2 + x) (x^2 + x + 1)^2 (7)} + 1

x^11 = {(x^2 + x) (x^2 + x +1) ((x^2 + x)^2 + (x^2 + x + 1)^3) (11)} + 1


x^13 = {(x^2 + x) (x^2 + x + 1)^2 (2 (x^2 + x)^2 + (x^2 + x + 1)^3) (13)} + 1

I have Mathematica notebooks with the computations for x^7,x^11, and x^13, verifying this up to x=100 and each series gap between successive members follows the pattern. However at x^17, I ran out of steam and cognitive resources. Prime factoring the [[gap term] -1] is how I started looking at the other series, and there is some frustratingly periodic behavior in the prime factors. Twin prime exponents are also connected in some way, but I cannot figure it out.

I have no idea what this means, if anything at all.

I emailed a Mathematician who started a relatively successful website devoted to Math and Integers about this, and this was his reply. I am not sure if this explains my original question, as Pascal's Triangle would account for the P factor in the gap, but the P is only one part of the functions.

I am also a little confused on the Maple program that was used, as you can see below, the outputs the Mathematician returned to me use composite exponents that increase (x^6,x^5, etc) whereas the initial functions only use x^2 and x^3. Can someone explain what he is talking about?


[I]Dear Gordon, Thanks for that message.
What makes this work is an old theorem of
Lucas, that says that if p is a prime, every term in
the p-th row of Pascal's triangle (except the 1's at the ends) is
divisible by p.

To do this algebraically, - which is what you are doing -
we can take (x+1)^p, subtract x^p and 1, and then
what is left will be divisible by p and we can try to
factor it.

Here I will do that in Maple:

> f:=n->lprint(factor((x+1)^n-x^n-1));

> f(2);
2*x

> f(3);
3*x*(1+x)

> f(5);
5*x*(1+x)*(x^2+x+1)

> f(7);
7*x*(1+x)*(x^2+x+1)^2

> f(11);
11*x*(1+x)*(x^2+x+1)*(x^6+3*x^5+7*x^4+9*x^3+7*x^2+3*x+1)
> f(13);
13*x*(1+x)*(x^6+3*x^5+8*x^4+11*x^3+8*x^2+3*x+1)*(x^2+x+1)^2
> f(17);
17*x*(1+x)*(x^2+x+1)*(x^12+6*x^11+26*x^10+75*x^9+156*x^8+240*x^7+277*x^ 6+240*x^5+156*x^4+75*x^3+26*x^2+6*x+1)

> f(19);
19*x*(1+x)*(x^12+6*x^11+28*x^10+85*x^9+184*x^8+292*x^7+341*x^6+292*x^5+ 184*x^4+85*x^3+28*x^2+6*x+1)*(x^2+x+1)^2

> f(23);
23*x*(1+x)*(x^2+x+1)*(x^18+9*x^17+57*x^16+252*x^15+836*x^14+2156*x^13+4 423*x^12+7324*x^11+9880*x^10+10911*x^9+9880*x^8+7324*x^7+4423*x^6+2156* x^5+836*x^4+252*x^3+57*x^2+9*x+1)

Further simplifications get trickier, as you noticed.


Thanks for your prompt and thoughtful reply. I will look into the book, it sounds very interesting. I was looking over the Maple outputs, and comparing it to the initials functions to see where they went wrong, because there seemed to be a discrepancy after x^7. I know you are probably busy, but I can't figure out why the Maple output has x^n (n from 1 to 6), when the erroneous function is merely squares and cubes.


I am just getting more confused the more people I ask.

It's not just Pascal's Triangle that evidences such periodic behavior modulo p. All manner of recursively defined triangles demonstrate such behavior, from the Stirling 1 and 2 Triangles to the Eulerian Triangle to the Bernoulli Triangle and beyond.

I don't know the theorem that would apply, but the patterns are more than apparent with a little trial and error.

- AC

P.S. Ramsey 2879, are you familiar with a general theorem that generalizes Lucas' therorem re: Pascal's Triangle to other recursively defined triangles?

Is it really ALL? That makes this far more interesting. Mind linking a paper or something?